On a certain class of curves arising in a free boundary problem for Stokes flows (Q1921806)

A compactness property is studied for a sequence \(\Omega_n \in \mathbb{R}^2\) of bounded simply connected domains under the assumption that there is a uniform control of the perimeters \(P(\Omega_n) \leq L\), and the first variation \(I_n (\Phi) = \int_{\partial \Omega_n} e(s) \Phi'e (s)ds\) is bounded uniformly in \(n\) in the following sense: \(|I_n (\Phi) |\leq E|D \Phi |_{L^2 (\Omega_n)}\) for any \(\Phi\in D(\mathbb{R}^2)\), \(\text{div} \Phi = 0\). Here \(e\) is the unit tangential vector on \(\partial \Omega_n\), \(D (\Phi)_{i,j} = {\partial \Phi_i \over \partial x_j} + {\partial \Phi_j \over\partial x_i}\). A convergence \(\Omega_n \to G\) to an open set \(G\) is defined to state that \(G\) is a strongly connected domain whenever \(G\neq \emptyset\). If \(G = \emptyset\), then \(\Omega_n\) shrink into a point. It follows that a liquid drop with bounded dissipation does not fragment into parts.